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Andrei Seymour-Howell works in computational number theory, particularly automorphic forms and trace formulas. His 2022 paper develops an algorithm to compute and rigorously verify Maass cusp forms for specified groups of squarefree level and trivial character. It combines a Selberg trace formula involving Hecke operators with linear algebra to isolate spectral information.^[ref]

The distinction between computation and certification is central. A decimal approximation to an eigenvalue suggests where to look; a rigorous verification must justify that the computed information corresponds to an actual mathematical object, with controlled error. As a schematic illustration, a proved enclosure $9.99\le \lambda \le 10.01$ conveys a different kind of information from a program merely printing 10.00. The numbers here illustrate the distinction and are not a reported eigenvalue. The paper applies its verified computations to gather evidence about further conjectures; verified examples still do not prove a conjecture quantified over all forms.^[ref]

An automorphic form is a function with arithmetic symmetries, together with analytic regularity and growth conditions. In a classical example, a modular form of integer weight $k$ on the modular group is a holomorphic function on the upper half-plane satisfyingand a holomorphy condition at the cusp. The transformation rule expresses compatibility with a group action, rather than ordinary periodicity alone: translation gives $f(z+1)=f(z)$, while inversion gives $f(-1/z)=z^{k}f(z)$.

The broader theory also permits nonholomorphic forms and groups beyond ${\mathrm{SL}}_2$. In the adelic formulation for ${\mathrm{GL}}_2$, automorphic forms are smooth, arithmetically invariant functions satisfying moderate growth and finiteness conditions under compact-group translations and central differential operators. The classical modular forms embed in this larger space. This connects number theory with representation theory and harmonic analysis.^[ref]

Ilya Piatetski-Shapiro was a mathematician whose research linked analytic number theory, representation theory and algebraic geometry. His early work included the distribution of prime numbers in special sequences and uniqueness questions for trigonometric series. His later work on automorphic forms and representations of Lie groups helped connect arithmetic questions with harmonic analysis.^[ref] These connections are powerful because the same object can carry both arithmetic information and a rich set of symmetries: methods for studying its transformations can then reveal properties of its arithmetic coefficients.

Identity and discipline: en-exact-sciences.tau.ac.il/math/in-memoriam

In the weight-zero convention, a Maass form for a discrete subgroup $\Gamma \subseteq {\mathrm{SL}}_2(\mathbb{R})$ is a smooth function on the upper half-plane, invariant under $z\mapsto (az+b)/(cz+d)$, satisfying a hyperbolic Laplacian eigenvalue equation and specified growth conditions at the cusps:Additional conditions specify a cuspidal or square-integrable class. The sign of $\Delta$ is a convention and must be stated.^[ref]

The constant function $f=1$ is an invariant eigenfunction with $\lambda =0$ and moderate growth, but is not cuspidal: its average along a horizontal period is nonzero. A Maass cusp form has zero such constant term at each cusp. The eigenfunction condition allows nonholomorphic automorphic forms, connecting arithmetic symmetries with spectral geometry.

Matthew Emerton is a mathematician working in number theory whose work connects modular forms, representation theory and arithmetic geometry. His authored account of classical modular forms as automorphic forms gives a precise translation between a weight-$k$ holomorphic function on the upper half-plane and an arithmetically invariant function on an adelic group.^[ref]

The translation converts a transformation rule into invariance and differentiation conditions. In particular, being holomorphic becomes annihilation by a suitable differential operator, while the weight becomes a character under rotations. This places classical modular forms in the same representation-theoretic language as more general smooth automorphic forms. His author page provides the note and identifies his research area as number theory.^[ref]

Paul Garrett works on automorphic forms, L-functions, representation theory, harmonic analysis and number theory.^[ref] These subjects bring together functions constrained by symmetry, linear actions of groups and analytic descriptions of arithmetic. His research and expository materials explore the interaction of those viewpoints.

Identity source: cse.umn.edu/math/paul-garrett

The Rankin–Selberg method relates arithmetic Dirichlet series to integrals of automorphic forms against Eisenstein series. Expanding the Eisenstein series as a sum over a coset space lets one replace an integral over a fundamental domain by an integral over a simpler region. This step is called unfolding.

For a weight-$k$ level-one cusp form $f(\tau )={\sum }_{n\ge 1}{a}_n{e}^{2\pi in\tau }$, the primitive real-analytic Eisenstein series $E(\tau ,w)$ gives, initially for large $\mathrm{Re}w$,Unfolding produces the strip $0\le x<1$, integration in $x$ removes unequal Fourier modes, and integration in $y$ supplies the gamma function. The Rankin–Selberg integral for holomorphic cusp forms makes this link precise for two forms. Analytic properties of the Eisenstein series then control the coefficient Dirichlet series. For real $w>1$, positivity also proves convergence through positive unfolding of a cusp-form square; analytic continuation alone would not justify convergence of the original sum.

A smooth automorphic form is an automorphic form whose archimedean variables are infinitely differentiable, without necessarily being holomorphic. Conventions differ over whether compact-group finiteness is included or one takes a larger smooth representation space. In the classical finite-vector convention on $G={\mathrm{SL}}_2(\mathbb{R})$, one asks for left invariance under an arithmetic subgroup $\Gamma$, moderate growth, finite-dimensional span under right translations by $K=\mathrm{SO}(2)$, and finiteness under the centre of the universal enveloping algebra.^[ref]

The constant function on $\Gamma \backslash G$ is an example: it is smooth, bounded, $K$-invariant and annihilated by positive-order invariant differentiation. More interesting examples come from Maass forms, whose Laplacian eigenfunction condition replaces holomorphy. Smoothness is therefore a regularity requirement, not a claim that the Cauchy-Riemann equations hold.